1,3

From Chai Wah Wu, Mar 02 2024: (Start)

Theorem: k is a term if and only if k is 0, 1, 3, 6 or of the form 2^(6*m+1), 5^(6*m+1), or 7^(3*m+1), m >= 0.

Proof: 0, 1, 3, 6 can be shown to be terms by direct computation. Since 1 <= A010888(k) <= 9 for k > 0, all terms > 0 are of the form q^m, where 2 <= q <= 9. This implies that all terms > 1 are either 6^m or of the form p^m, where p is a prime <= 7. If k = 6^m for m > 1, then k is divisible by 9 and A010888(k) = 9, whereas A052410(k) = 6 and thus k is not a term. Similarly, if k = 3^m for m > 1, then k is divisible by 9 and A010888(k) = 9 whereas A052410(k) = 3 and thus k is not a term. Lastly, for p = 2, 5 or 7, A010888(p^m) = 1 + ((p^m-1) mod 9), and since p^m and 9 are coprime, this implies that A010888(p^m) = p^m mod 9 whereas A052410(p^m) = p, thus m must satisfy p^m mod 9 = p mod 9. Since 6, 6, 3 are the multiplicative orders of 2, 5, 7 modulo 9 respectively, implying 2^6 == 5^6 == 7^3 == 1 (mod 9), the result follows.

(End)

A052409[n_]:=GCD@@Last/@FactorInteger[n]; A010888[n_]:=If[n==0, 0, n-9 Floor[(n-1)/9]]; a={}; kmax = 10^9; For[k=0, k<=kmax, k++, If[k^(1/A052409[k])==A010888[k], AppendTo[a, k]]]; a

(Python)

from itertools import count, islice

from math import gcd

from sympy import factorint, integer_nthroot

def A370688_gen(startvalue=0): # generator of terms >= startvalue

if startvalue <=0: yield 0

if startvalue <=1: yield 1

for k in count(max(startvalue, 2)):

r = 1 + (k - 1) % 9

if r>1:

kmin, kmax = 0, 1

while r**kmax <= k:

kmax <<= 1

while True:

kmid = kmax+kmin>>1

if r**kmid > k:

kmax = kmid

else:

kmin = kmid

if kmax-kmin <= 1:

break

if r**kmin==k:

m = integer_nthroot(k, gcd(*factorint(k).values()))[0]

if m == r:

yield k

A370688_list = list(islice(A370688_gen(), 10)) # Chai Wah Wu, Mar 02 2024

(Python)

# faster program based on theorem

from itertools import islice

def A370688_gen(): # generator of terms

kmax, mlist, dlist = 10, [7, 7, 4], [6, 6, 3]

yield from (0, 1, 2, 3, 5, 6, 7)

while True:

klist = []

for i, p in enumerate((2, 5, 7)):

while (k:=p**mlist[i]) <= kmax:

klist.append(k)

mlist[i] += dlist[i]

yield from sorted(klist)

kmax *= 10

A370688_list = list(islice(A370688_gen(), 10)) # Chai Wah Wu, Mar 02 2024

Cf. A010888, A052409, A052410.

Sequence in context: A357263 A067077 A389195 * A357132 A067183 A269670

Adjacent sequences: A370685 A370686 A370687 * A370689 A370690 A370691

nonn,base

Stefano Spezia, Feb 27 2024

a(18)-a(27) from Chai Wah Wu, Mar 02 2024

approved